In this paper, we study the global dynamics of three higher-order exponential systems of rational difference equations. This suggested work considerably extends and improves some existing results in the literature.

In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation $ n = 0, 1, ..., $ where the initial conditions and$ x_{0}\ $are arbitrary real numbers and the values $ a, \ b, \ c\ $and$ \;d $ are defined as positive real numbers.

In this study, we examined a (2+1)-dimensional generalized breaking soliton system (GBSS) using both analytical and numerical methods. By applying a generalized direct algebraic method, we derived exact solutions that displayed a variety of solitary and periodic wave patterns. These solutions illuminated the interplay between nonlinearity and dispersion in several physical contexts, including fluid dynamics, plasma physics, and nonlinear optics. In addition, we developed a robust numerical scheme employing an adaptive moving mesh technique based on the MMPDE5 framework. Stability and error analyses confirmed that this method concentrated grid points around steep gradients, achieved second-order spatial convergence, and enhanced computational efficiency. By comparing numerical and exact solutions, we provided more profound insights into GBSS dynamics and facilitated future investigations of complex, multidimensional, and nonlinear wave phenomena.

In this article, we investigate the weak higher order nonlinear solitonic pressure waves in elastic, incompressible, nonviscous fluid-filled tubes. The higher order Korteweg–de Vries equation has been developed from the perturbed nonlinear equation. The dependence of higher-order solitonic pressure, higher order fluid velocity, and higher order solitonic energy on the nonlinear coefficient and the perturbation parameter has been established. The presented solutions will be more beneficial in the several disciplines of blood flow and many other related fields of applied science.

In this paper, we consider two nonlinear models arising in mathematical physics, namely, the Landau–Ginzburg–Higgs (LGH) equation and the nonlinear dispersive modified Benjamin–Bona (DMBBM) equation. The LGH model describes the exchange between mid-latitude Rossby and equatorial waves, as well as nonlinear waves with long-range and weak scattering interactions between tropical tropospheres and mid-latitude. The DMBBM model explains surface wave propagation estimates in a nonlinear dispersive medium. We employed He’s semi-inverse approach in order to solve these models. Specifically, we present hyperbolic wave solutions. The proposed method is straightforward, reliable, and effective, and its potential for use in solving additional partial differential equations in applied research is encouraging. Appropriate values for the parameters are taken into consideration when simulating certain 2D and 3D graphs that correspond to select solutions using Matlab software.

Despite substantial research into nonlinear Schrödinger equation equations, present techniques frequently tackle resonant nonlinear dynamics and stochastic perturbations separately, limiting comprehension of their combined impact on wave propagation. In this paper, we address this gap by creating a unified computational framework for the stochastic resonance nonlinear Schrödinger equation, a model that captures the interaction of nonlinear wave dynamics, resonant interactions, and ambient noise. The suggested unified techniques allow for the creation of novel stochastic wave structures. These solutions are essential for accurately capturing the influence of environmental noise and random perturbations on nonlinear wave dynamics, thereby providing a more realistic description of resonance, stability, and energy transfer mechanisms. They also offer deeper insight into noise-induced phenomena, such as stochastic resonance, modulation instability, and the formation of complex wave structures, which cannot be explained by deterministic models alone. We further demonstrate how variations in the physical parameters and the stochastic noise term systematically influence the behavior of the proposed solutions. The proposed techniques provide a generalizable framework for future analytical and computational studies of stochastic nonlinear partial differential equations.